We have derived solutions for the classical FL equation (REF ) and nonlocal FL equation (REF )
from bilinear approach.
We introduced new double Wronskian expressions (REF ) that are different from
those of the AKNS hierarchy, the KN equation and the Chen-Lee-Liu equation
(cf.[1]}, [2]}, [3]}).
The assumption (REF ) with a general \(A\)  for Wronskian entries
and the reduction technique enable us to have a full profile
for the solutions of the FL equations.
One- and two-soliton solutions were illustrated based on analysis in detail.
It is notable that the FL equation (REF ) also allows solutions related to real discrete eigenvalues.
They exhibit (multi-)periodic behavior for distinct eigenvalues and algebraic decayed solitary waves
for those eigenvalues with multiplicity two. The later case was not
found before in the analytic approaches (e.g. [4]}, [5]}, [6]})
that are based on analyzing analytic domains of wave functions.
